Let \(A\)  be a set, and let \(A^{[\bullet ]}\)  denote the simplicial set with \((A^{[\bullet ]})_n = A^{n+1}\) , and whose \(i\) -th face map forgets the \(i\) -th entry of a tuple.
In other words, \(A^{[\bullet ]}\)  is the Čech nerve of the collapse map \(A \rightarrow *\)  in \({{S}\mathrm {et}}\) .
Let \({\operatorname{Ex}}\colon {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\rightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  be the right adjoint to the simplicial subdivision functor (for more background, see Appendix  and [1]}).
There exists a morphism of simplicial sets,
\(\sigma _A \colon A^{[\bullet ]} \longrightarrow {\operatorname{Ex}}N(sA)\,,\) 
